Extinction and Survival in an Interval-Activation Frog Model on \mathbb{Z} with Random Survival Parameters and Symmetric Random Walks

We study an interval-activation frog model on $\mathbb Z$ with i.i.d.\ initial numbers of frogs $(\eta_x)_{x\in\mathbb Z}$, satisfying $0<\mathbb{E}[\eta_0]<\infty$. Frogs at the origin are initially active and all others are sleeping. Each frog performs a symmetric integer-valued random walk and has a random lifetime $L$ determined by an i.i.d.\ survival parameter $\pi\in(0,1)$, with $\mathbb{P}(L\ge k\mid \pi=p)=p^k$. Every jump activates all sleeping frogs at the integer sites between its endpoints. Let $D^\to$ denote the maximal rightward displacement of a single frog before death. We derive survival and extinction criteria from the tail behavior of $D^\to$. If $\mathbb{P}(|\xi_1|\ge n)\sim n^{-\alpha}L_\xi(n)$, with $L_\xi$ slowly varying, then survival holds with positive probability for $0<\alpha<1$, while for $\alpha=1$ both survival and almost sure extinction may occur. For $1<\alpha<2$, assume $\mathbb{P}(|\xi_1|>n)\sim c_\xi n^{-\alpha}$; in the finite-variance case assume $\mathbb{E}[\xi_1]=0$ and $\operatorname{Var}(\xi_1)=\sigma^2\in(0,\infty)$. Setting $r=\alpha$ in the stable case and $r=2$ in the finite-variance case, if the law of $\pi$ has density $f_\pi(u)\sim(1-u)^{\beta-1}\ell((1-u)^{-1})$ as $u\uparrow1$, then, for $0<\beta<1$, $n\mathbb{P}(D^\to\ge n)\sim C_\beta n^{1-r\beta}\ell(n^r)$, with explicit $C_\beta$. Hence the sharp off-critical threshold is $\beta_c=1/r$: survival holds for $\beta<1/r$, extinction holds almost surely for $\beta>1/r$, and explicit sufficient conditions on the critical line leave a factor-four gap.